The Jordan-Holder theorem says that any chain of subobjects of a finite length object can be refined to a composition series, and that any composition series has the same length. This theorem holds for any abelian category, and a notable example is the case of modules over some ring. While I do not need an example of the usefulness of JH theorem in the context of modules, I would like to ask: > What are applications of the JH theorem in a general abelian category, > which is not (or not easily proven to be) a category of modules? [This question was originally asked on [mathstackexchange][1], but it received upvotes with no answers. I think it is more likely it will get interesting answers on this site.] [1]: https://math.stackexchange.com/questions/3378844/applications-of-jordan-holder-theorem-in-an-abelian-category